Chapter 7 Mechanical behaviour of materials 7.1 Mechanical testing procedures The load-elongation curves for both polycrystalline 7.1.1 Introduction mild steel and copper are shown in Figures 7.1a and 7.Ib, The corresponding stress (load per unit area, Real crystals, however carefully prepared, contain P/A)versus strain( change in length per unit length, lattice imperfections which profoundly affect those dl/1) curves may be obtained knowing the dimensions properties sensitive to structure. Careful examination of the test piece. At low stresses the deformation is of the mechanical behaviour of materials can give elastic, reversible and obeys Hooke's law with stress information on the nature of these atomic defects. linearly proportional to strain The proportionality con- In some branches of industry the common mechan stant connecting stress and strain is known as the cal tests, such as tensile, hardness, impact, creep and elastic modulus and may be either (a)the elastic or fatigue tests, may be used, not to study the'defect Young's modulus, E, (b) the rigidity or shear modulus duced against a standard specification. Whatever its the strain is tensile, shear or hydrostatic compressive, purpose, the mechanical test is of importance in the respectively. soissons ratio v, the ratio of lateral con- development of both materials science and engineer- tractions to longitudinal extension in uniaxial tension It is different machines for performing the tests are in gen- are related according to eral use. This is because it is often necessary to know E E the effect of temperature and strain rate at different 2(1-2u) μ=2(1+) levels of stress depending on the material being tested. 3K+ Consequently, no attempt is made here to describe the (71) details of the various testing machines. The elements but for impure iron and low carbon steels the onset of plastic deformation is denoted by a sudden drop in 7.1.2 The tensile test load indicating both an upper and lower yield point. This yielding behaviour is characteristic of many met In a tensile test the ends of a test piece are fixed into als, particularly those with bcc structure containing grips, one of which is attached to the load-measuring small amounts of solute element(see Section 7. 4.6) evice on the tensile machine and the other to the For materials not showing a sharp yield point, a co straining device. The strain is usually applied by means ventional definition of the beginning of plastic filow of a motor-driven crosshead and the elongation of the 0. 1 %o proof stress, in which a line is drawn parallel the specimen is indicated by its relative movement The load necessary to cause this elongation may be Load relaxations are obtained only on'hard'beam obtained from the elastic deflection of either a beam olanyi-type machines where the beam deflection is small or proving ring, which may be measured by usil hydraulic, optical or electromechanical methods. The hich the load-measuring device is a soft spring, rapid load last method (where there is a change in the resistance variations are not recorded because the extension re too large, while in dead-loading machines no lo of strain gauges attached to the beam) is, of course, relaxations are possible. In these latter machines sudden easily adapted into a system for autographically record lding will show as merely an extension under constant ing the load-elongation curve
Chapter 7 Mechanical behaviour of materials 7.1 Mechanical testing procedures 7.1.1 Introduction Real crystals, however carefully prepared, contain lattice imperfections which profoundly affect those properties sensitive to structure. Careful examination of the mechanical behaviour of materials can give information on the nature of these atomic defects. In some branches of industry the common mechanical tests, such as tensile, hardness, impact, creep and fatigue tests, may be used, not to study the 'defect state' but to check the quality of the product produced against a standard specification. Whatever its purpose, the mechanical test is of importance in the development of both materials science and engineering properties. It is inevitable that a large number of different machines for performing the tests are in general use. This is because it is often necessary to know the effect of temperature and strain rate at different levels of stress depending on the material being tested. Consequently, no attempt is made here to describe the details of the various testing machines. The elements of the various tests are outlined below. 7.1.2 The tensile test In a tensile test the ends of a test piece are fixed into grips, one of which is attached to the load-measuring device on the tensile machine and the other to the straining device. The strain is usually applied by means of a motor-driven crosshead and the elongation of the specimen is indicated by its relative movement. The load necessary to cause this elongation may be obtained from the elastic deflection of either a beam or proving ring, which may be measured by using hydraulic, optical or electromechanical methods. The last method (where there is a change in the resistance of strain gauges attached to the beam) is, of course, easily adapted into a system for autographically recording the load-elongation curve. The load-elongation curves for both polycrystalline mild steel and copper are shown in Figures 7.1a and 7.lb. The corresponding stress (load per unit area, P/A) versus strain (change in length per unit length, dl/l) curves may be obtained knowing the dimensions of the test piece. At low stresses the deformation is elastic, reversible and obeys Hooke's law with stress linearly proportional to strain. The proportionality constant connecting stress and strain is known as the elastic modulus and may be either (a)the elastic or Young's modulus, E, (b) the rigidity or shear modulus /z, or (c) the bulk modulus K, depending on whether the strain is tensile, shear or hydrostatic compressive, respectively. Young's modulus, bulk modulus, shear modulus and Poisson's ratio v, the ratio of lateral contractions to longitudinal extension in uniaxial tension, are related according to E E 9K/.t K= 2(1-2v)' /.t= 2(1+v)' E= 3K+/z (7.1) In general, the elastic limit is an ill-defined stress, but for impure iron and low carbon steels the onset of plastic deformation is denoted by a sudden drop in load indicating both an upper and lower yield point. 1 This yielding behaviour is characteristic of many metals, particularly those with bcc structure containing small amounts of solute element (see Section 7.4.6). For materials not showing a sharp yield point, a conventional definition of the beginning of plastic flow is the 0.1% proof stress, in which a line is drawn parallel 1Load relaxations are obtained only on 'hard' beam Polanyi-type machines where the beam deflection is small over the working load range. With 'soft' machines, those in which the load-measuring device is a soft spring, rapid load variations are not recorded because the extensions required are too large, while in dead-loading machines no load relaxations are possible. In these latter machines sudden yielding will show as merely an extension under constant load
198 Modern Physical Metallurgy and Materials Engineering Upper Unirradiated td point re (19x20° neutrons Luders strain per cm2) 100 Uniform elongatIon 60 Figure 7.1 Stress-elongation curves for (a)impure iron, (b)copper, (c) ductile-brittle transition in mild steel(after Churchman, Morford and Cottrell, 1957 to the elastic portion of the stress-strain curve from than the strain to fracture measured along the gauge For control purposes the tensile test gives valuable length the point of 0. 1 strain True stress-true strain curves are often plotted to information on the tensile strength(TS= maximum show the work hardening and strain behaviour at large ad/original area)and ductility(percentage reduction strains. The true stress o is the load P divided by the in area or percentage elongation) of the material. When area A of the specimen at that particular stage of strain it is used as a research technique, however, the exact and the total true strain in deforming from initial length ape and fine details of the curve, in addition to the lo to length Ii is e=Ji(dl/I)=In(11/lo). The true way in which the yield stress and fracture stress vary stress-strain curves often fit the Ludwig relation a with temperature, alloying additions and grain size, are ke" where n is a work-hardening coefficient 0. 1-0 robably of greater significance and k the strength coefficient. Plastic instability, or The increase in stress from the initial yield up to the necking, occurs when an increase in strain produces TS indicates that the specimen hardens during defor- no increase in load supported by the specimen, i.e mation (i.e. work-hardens). On straining beyond the dP=0, and hence since P= aA. then TS the metal still continues to work-harden, bi ll to compensate for the reduction in ross-sectional area of the test piece. The deforma- defines the instability condition. During deformation, tion then becomes unstable, such that as a localized the specimen volume is essentially constant (i.e. dv= region of the gauge length strains more than the rest, 0)and from it cannot harden sufficiently to raise the stress for fur- ther deformation in this region above that to cause dv =d(IA)= AdI+ldA=0 further strain elsewhere. A neck then forms in the we obtain lis region until fracture. Under these conditions. the da d/ eduction in area(Ao- At)/Ao where Ao and A, are the initial and final areas of the neck gives a mea- Thus, necking at a strain at which the slope sure of the localized strain, and is a better indication of the true stress-true strain curve equals the true
198 Modern Physical Metallurgy and Materials Engineering "~ 200 Fracture Z ~- Lower y~etd point i ' --<.. I00- Luders strain C .... 1 t 1 1 1 ..... __ (a) Max tmum stress 200 I00 Totat etonga ion 1 . L- 0 in -1 10 Etongation ~ % (b) 6050 ~'f~ radiated 40 i Irradiated --, S 30 2O ,o ~ ....... '40 0 40 8'o Temperature ~ ~ (c) Figure 7.1 Stress-elongation curves for (a) impure iron, (b) copper, (c) ductile-brittle transition in mild steel (after Churchman, Mogford and Cottrell, 195 7). to the elastic portion of the stress-strain curve from the point of 0.1% strain. For control purposes the tensile test gives valuable information on the tensile strength (TS- maximum load/original area) and ductility (percentage reduction in area or percentage elongation) of the material. When it is used as a research technique, however, the exact shape and fine details of the curve, in addition to the way in which the yield stress and fracture stress vary with temperature, alloying additions and grain size, are probably of greater significance. The increase in stress from the initial yield up to the TS indicates that the specimen hardens during deformation (i.e. work-hardens). On straining beyond the TS the metal still continues to work-harden, but at a rate too small to compensate for the reduction in cross-sectional area of the test piece. The deformation then becomes unstable, such that as a localized region of the gauge length strains more than the rest, it cannot harden sufficiently to raise the stress for further deformation in this region above that to cause further strain elsewhere. A neck then forms in the gauge length, and further deformation is confined to this region until fracture. Under these conditions, the reduction in area (A0- A l)/Ao where A0 and A l are the initial and final areas of the neck gives a measure of the localized strain, and is a better indication than the strain to fracture measured along the gauge length. True stress-true strain curves are often plotted to show the work hardening and strain behaviour at large strains. The true stress o is the load P divided by the area A of the specimen at that particular stage of strain and the total true strain in deforming from initial length Io to length ll is e---- f/o' (dl/l)= ln(l~/lo). The true stress-strain curves often fit the Ludwig relation a = ke" where n is a work-hardening coefficient ~0.1-0.5 and k the strength coefficient. Plastic instability, or necking, occurs when an increase in strain produces no increase m load supported by the specimen, i.e. dP = 0, and hence since P -- oA, then dP = Ado + odA = 0 defines the instability condition. During deformation, the specimen volume is essentially constant (i.e. dV = 0) and from dV = d(/a) = Adl + ldA -- 0 we obtain do dA dl a -- A -- 1 --dE (7.2) Thus, necking occurs at a strain at which the slope of the true stress-true strain curve equals the true
its deformation behaviour. The hardness tester forces small sphere, pyramid or cone into the surface of the metals by means of a known applied load, and the hardness number( Brinell or Vickers diamond pyramid) then obtained from the diameter of the impres Instability The hardness may be related to the yield or tensile trength of the metal, since during the indentation, the Nominal strain En material around the impression is plastically deformed to a certain percentage strain. The vickers hardness Figure 7.2 Considere's construction number(VPN) is defined as the load divided by the pyramidal area of the indentation, in kgf/mm, and is about three times the yield stress for materials which stress at that strain, i.e. do/dE= o. Alternatively, since do not work harden appreciably. The Brinell hardness ke"= a= do/de= nke- then e=n and necking number(BHN) is defined as the stress P/A, in kgf/mm occurs when the true strain equals the strain-hardening where P is the load and A the surface area of the exponent. The instability condition may also be spherical cap forming the indentation. Thus expressed in terms of the conventional(nominal strain) dl/o do l BHN=P/D2)(-(-(d/D)21 /2) dl/I where d and d are the indentation and indentor diam =+(1+En)=a eters respectively. For consistent results the ratio d/D (7.3) should be maintained constant and small. Under these conditions soft materials have similar values of BHN which allows the instability point to be located using and VPN. Hardness testing is of importance in both Considere's construction(see Figure 7. 2), by plotting control work and research, especially where informa the true stress against nominal strain and drawing the tion on brittle materials at elevated temperatures tangent to the curve from En=-I on the strain axis. required The point of contact is the instability stress and the tensile strength is a/(1+En) Tensile specimens can also give information on the 7. 1. 4 Impact testing pe of fracture exhibited. Usually in polycrystalline A material may have a high tensile strength and yet metals transgranular fractures occur (i.e. the fracture be unsuitable for shock loading conditions. To deter urface cuts through the grains )and the cup and cone mine this the impact resistance is usually measured by type of fracture is extremely common in really duc- means of the notched or un-notched Izod or Charpy tile metals such as copper. In this, the fracture starts impact test. In this test a load swings from a given at the centre of the necked portion of the test piece height to strike the specimen, and the energy dissi s measured. The test is parti xis, so forming the cup,, but then, as it nears the ularly useful in showing the decrease in ductility and outer surface, it turns into a ' by fracturing along impact strength of materials of bcc structure at mod surface at about 45 to the tensile axis. In detail erately low temperatures. For example, carbon steels the itself consists of many irregular surfaces at have a relatively high ductile-brittle transition tem- about 45to the tensile axis, which gives the fracture a perature(Figure 7. Ic)and, consequently, they may be fibrous appearance. Cleavage is also a fairly common sed with safety at sub-zero temperatures only if the type of transgranular fracture, particularly in materi- transition temperature is lowered by suitable alloy als of bcc structure when tested at low temperatures ons or by refining the grain size. No The fracture surface follows certain crystal planes (e.g. increasing importance is given to defining a fracture [100) planes), as is shown by the grains revealing toughness parameter K for an alloy, since many alloys large bright facets, but the surface also arpels where critical stress, propagate; K, defines the critical com- contain small cracks which, when subjected to some cleavage planes have been tom apart. Intercrystallir fractures sometimes occur, often without appreciable discussed more fully in Chapter 8 deformation. This type of fracture is usually caused by a brittle second phase precipitating out around the 7.1.5 Creep testing grain boundaries, as shown by copper containing bis muth or antimony Creep is defined as plastic flow under constant stress and although the majority of tests are carried out under 7.1.3 Indentation hardness testing constant load conditions, equipment is available for reducing the loading during the test to compensate The hardness of a metal defined as the resistance to for the small reduction in cross-section of the spec- penetration,gives a conveniently rapid indication of imen. At relatively high temperatures creep appears to
Mechanical behaviour of materials 199 / )l/ : .Instobility / ff'y i/strain -1 0 Nominal strain E n Figure 7.2 Considbre's construction. stress at that strain, i.e. do"/de -- o.. Alternatively, since ke" = o. = dot/de = nke "-I then e = n and necking occurs when the true strain equals the strain-hardening exponent. The instability condition may also be expressed in terms of the conventional (nominal strain) do. do. de,, de de,, de do. (dl/lo) do. 1 de,, dl/)i = de, l0 do -- ~(1 + e,,) : o. (7.3) den which allows the instability point to be located using Consid6re's construction (see Figure 7.2), by plotting the true stress against nominal strain and drawing the tangent to the curve from e,, = -1 on the strain axis. The point of contact is the instability stress and the tensile strength is o./(1 + e,, ). Tensile specimens can also give information on the type of fracture exhibited. Usually in polycrystalline metals transgranular fractures occur (i.e. the fracture surface cuts through the grains) and the 'cup and cone' type of fracture is extremely common in really ductile metals such as copper. In this, the fracture starts at the centre of the necked portion of the test piece and at first grows roughly perpendicular to the tensile axis, so forming the 'cup', but then, as it nears the outer surface, it turns into a 'cone' by fracturing along a surface at about 45 ~ to the tensile axis. In detail the 'cup' itself consists of many irregular surfaces at about 45 ~ to the tensile axis, which gives the fracture a fibrous appearance. Cleavage is also a fairly common type of transgranular fracture, particularly in materials of bcc structure when tested at low temperatures. The fracture surface follows certain crystal planes (e.g. {100} planes), as is shown by the grains revealing large bright facets, but the surface also appears granular with 'river lines' running across the facets where cleavage planes have been torn apart. Intercrystalline fractures sometimes occur, often without appreciable deformation. This type of fracture is usually caused by a brittle second phase precipitating out around the grain boundaries, as shown by copper containing bismuth or antimony. 7.1.3 Indentation hardness testing The hardness of a metal, defined as the resistance to penetration, gives a conveniently rapid indication of its deformation behaviour. The hardness tester forces a small sphere, pyramid or cone into the surface of the metals by means of a known applied load, and the hardness number (Brinell or Vickers diamond pyramid) is then obtained from the diameter of the impression. The hardness may be related to the yield or tensile strength of the metal, since during the indentation, the material around the impression is plastically deformed to a certain percentage strain. The Vickers hardness number (VPN) is defined as the load divided by the pyramidal area of the indentation, in kgf/mm 2, and is about three times the yield stress for materials which do not work harden appreciably. The Brinell hardness number (BHN) is defined as the stress P/A, in kgf/mm 2 where P is the load and A the surface area of the spherical cap forming the indentation. Thus BUN --- P/(---~D2)2 / {1 -[1 -(d/O)2] '/2} where d and D are the indentation and indentor diameters respectively. For consistent results the ratio diD should be maintained constant and small. Under these conditions soft materials have similar values of B HN and VPN. Hardness testing is of importance in both control work and research, especially where information on brittle materials at elevated temperatures is required. 7.1.4 Impact testing A material may have a high tensile strength and yet be unsuitable for shock loading conditions. To determine this the impact resistance is usually measured by means of the notched or un-notched Izod or Charpy impact test. In this test a load swings from a given height to strike the specimen, and the energy dissipated in the fracture is measured. The test is particularly useful in showing the decrease in ductility and impact strength of materials of bcc structure at moderately low temperatures. For example, carbon steels have a relatively high ductile-brittle transition temperature (Figure 7.1c) and, consequently, they may be used with safety at sub-zero temperatures only if the transition temperature is lowered by suitable alloying additions or by refining the grain size. Nowadays, increasing importance is given to defining a fracture toughness parameter Kc for an alloy, since many alloys contain small cracks which, when subjected to some critical stress, propagate; Kc defines the critical combination of stress and crack length. Brittle fracture is discussed more fully in Chapter 8. 7.1.5 Creep testing Creep is defined as plastic flow under constant stress, and although the majority of tests are carded out under constant load conditions, equipment is available for reducing the loading during the test to compensate for the small reduction in cross-section of the specimen. At relatively high temperatures creep appears to
200 Modern Pl Metallurgy and Materials Engineering tension or compression, but all involve the same prin ciple of subjecting the material to constant cycles of stress. To express the characteristics of the stress sys tem, three properties are usually quoted: these include (I)the maximum range of stress, (2)the mean stress, and(3)the time period for the stress cycle. Four dif ferent arrangements of the stress cycle are shown in Figure 7. 4, but the reverse and the repeated cycle tests (e.g. push-pull")are the most common, since they are L。wr,Low the easiest to achieve in the laboratory. and to subject them to tests using a different rang of stress, S, on each group of specimens. The num Figure 7.3 Typical creep curves. ber of stress cycles, N, endured by each specimen at a given stress level is recorded and plotted, as shown occur at all stress levels, but the creep rate increase in Figure 7.5. This S-N diagram indicates that some with increasing stress at a given temperature. For the metals can withstand indefinitely the application of a accurate assessment of creep properties, it is clear that large number of stress reversals, provided the applied special attention must be given to the maintenance of limit. for certain ferrous materials when they are used surement of the small dimensional changes involved. in the absence of corrosive conditions the assumption This latter precaution is ne of a safe working range of stress seems justified, bu rials a rise in temperature by a few tens of degrees for non-ferrous materials and for steels when they are used in corrosive conditions a definite endurance limit shows the characteristics of a typical creep curve and cannot be defined. Fatigue is discussed in more detail following the instantaneous strain caused by the sud divided into three stages, usually termed primary or 7. 1.7 Testing of ceramics transient creep, second or steady-state creep and ter- Direct tensile testing of ceramics is not generally tiary or accelerating creep. The characteristics of the favoured, mainly because of the extreme sensitivity of creep curve often vary, however, and the tertiary stage ceramics to surface flaws. First, it is difficult to apply of creep may be advanced or retarded if the tempera- a truly uniaxial tensile stress: mounting the specimen ture and stress at which the test is carried out is high in the machine grips can seriously damage the surface or low respectively(see Figure 7.3, curves b and c). and any bending of the specimen during the test will Creep is discussed more in Section 7.9 cause premature failure. Second, suitable waisted spec- imens with the necessary fine and flawless finish are 7.1.6 Fatigue testing expensive to produce. It is therefore common practice to use bend tests for engineering ceramics and glasses The fatigue phenomenon is concerned with the prema - (They have long been used for other non-ductile mate- ture fracture of metals under repeatedly applied low rials such as concretes and grey cast iron. In the three- stresses, and is of importance in many branches of and four-point bend methods portrayed in Figure 7.6,a engineering (e.g. aircraft structures ). Several differ- beam specimen is placed between rollers and carefully ent types of testing machines have been constructed loaded at a constant strain rate. The fexural strength in which the stress is applied by bending, torsion, at failure, calculated from the standard formulae, is Figure 7.4 Alternative forms of stress cycling:(a)reversed;(b)alternating(mean stress zero),(c) fluctuating and
200 Modern Physical Metallurgy and Materials Engineering s t c e~ u~ c/N,gh /', / i| Low r, Low d Ttrn~ Figure 7.3 Typical creep curves. occur at all stress levels, but the creep rate increase with increasing stress at a given temperature. For the accurate assessment of creep properties, it is clear that special attention must be given to the maintenance of the specimen at a constant temperature, and to the measurement of the small dimensional changes involved. This latter precaution is necessary, since in many materials a rise in temperature by a few tens of degrees is sufficient to double the creep rate. Figure 7.3, curve a, shows the characteristics of a typical creep curve and following the instantaneous strain caused by the sudden application of the load, the creep process may be divided into three stages, usually termed primary or transient creep, second or steady-state creep and tertiary or accelerating creep. The characteristics of the creep curve often vary, however, and the tertiary stage of creep may be advanced or retarded if the temperature and stress at which the test is carried out is high or low respectively (see Figure 7.3, curves b and c). Creep is discussed more fully in Section 7.9. 7.1.6 Fatigue testing The fatigue phenomenon is concerned with the premature fracture of metals under repeatedly applied low stresses, and is of importance in many branches of engineering (e.g. aircraft structures). Several different types of testing machines have been constructed in which the stress is applied by bending, torsion, tension or compression, but all involve the same principle of subjecting the material to constant cycles of stress. To express the characteristics of the stress system, three properties are usually quoted: these include (1) the maximum range of stress, (2) the mean stress, and (3) the time period for the stress cycle. Four different arrangements of the stress cycle are shown in Figure 7.4, but the reverse and the repeated cycle tests (e.g. 'push-pull') are the most common, since they are the easiest to achieve in the laboratory. The standard method of studying fatigue is to prepare a large number of specimens free from flaws, and to subject them to tests using a different range of stress, S, on each group of specimens. The number of stress cycles, N, endured by each specimen at a given stress level is recorded and plotted, as shown in Figure 7.5. This S-N diagram indicates that some metals can withstand indefinitely the application of a large number of stress reversals, provided the applied stress is below a limiting stress known as the endurance limit. For certain ferrous materials when they are used in the absence of corrosive conditions the assumption of a safe working range of stress seems justified, but for non-ferrous materials and for steels when they are used in corrosive conditions a definite endurance limit cannot be defined. Fatigue is discussed in more detail in Section 7.11. 7.1.7 Testing of ceramics Direct tensile testing of ceramics is not generally favoured, mainly because of the extreme sensitivity of ceramics to surface flaws. First, it is difficult to apply a truly uniaxial tensile stress: mounting the specimen in the machine grips can seriously damage the surface and any bending of the specimen during the test will cause premature failure. Second, suitable waisted specimens with the necessary fine and flawless finish are expensive to produce. It is therefore common practice to use bend tests for engineering ceramics and glasses. (They have long been used for other non-ductile materials such as concretes and grey cast iron.) In the threeand four-point bend methods portrayed in Figure 7.6, a beam specimen is placed between rollers and carefully loaded at a constant strain rate. The flexural strength at failure, calculated from the standard formulae, is (8) (b) (c) (d) Figure 7.4 Alternative forms of stress cycling: (a) reversed; (b) altet~ating (mean stress ~ zero), (c) fluctuating and (d) repeated
centre of an electric furnace heated by Sic elements 435 250 pe of the routine testing of graphite electrode samples and Carburized gives a useful indication of their ability to withstand accidental lateral impact during service in steel melting furnaces e Proof-testing is a long-established method of test g certain engineering components and structures In a typical proof test, each component is held at a Decarburized certain proof stress for a fixed period of time; load iron(0.004%C) ng and unloading conditions are standardized. In the ase of ceramics, it may involve bend-testing, inter- withstand the proof test are, in the simplest analysi udged to be sound and suitable for long-term service at the lower design stress. The underlying philosophy as been often questioned, not least because there is a Figure 7.5 S-N curve for carburized and decarburized iron risk that the proof test itself may cause incipient crack g. Nevertheless, proof-testing now has an important ole in the statistical control of strength in ceramics. 7.2 Elastic deformation 7. 2.1 Elastic deformation of metals 3 F(L-L It is well known that metals deform both elastically MoR= and plastically. Elastic deformation takes place at low stresses and has three main characteristics, namely 3-Point bend 4-Point b (1)it is reversible, (2)stress and strain are linearly proportional to each other according to Hooke's Law ons, MoR= modulus of and (3)it is usually small (i.e. <1%elastic strain) The stress at a point in a body is usually defined b=breadth of specimen, d by considering an infinitesimal cube surrounding that point and the forces applied to the faces of the cube by he surrounding material. These forces may be resolved known as the modulus of rupture(MoR)and expresses into components parallel to the cube edges and when the maximum tensile stress which develops on the con- divided by the area of a face give the nine stress vex face of the loaded beam. Strong ceramics, such as mponents shown in Figure 7.7. a given component silicon carbide and hot-pressed silicon nitride, have i is the force acting in the j-direction per unit ery high MoR values. The four-point loading method area of face normal to the i-direction. Clearly, when is often preferred because it subjects a greater volume i=j we have normal stress components(e.g.oux and area of the beam to stress and is therefore more which may be either tensile(conventionally positive) arching MoR values from four-point tests are often or compressive(negative), and when i+j(e.g oxy) substantially lower than those from three-point tests the stress components are shear. These shear stresses on the same material. Similarly, strength values tend exert couples on the cube and to prevent rotation of the to decrease as the specimen size is increased. To pro- cube the couples on opposite faces must balance and vide worthwhile data for quality control and design hence o =oi. Thus, stress has only six independent ctivities, close attention must be paid to strain rate components nd environment, and to the size, edge finish and sur. When a body is strained, small elements in that ace texture of the specimen. with oxide ceramics and body are displaced. If the initial position of an elem silica glasses, a high strain rate will give an appr iably higher flexural strength value than a low strain I'The nine components of stress ou form a second-rank rate, which leads to slow crack growth and delayed tensor usually written ction The bend test has also been adapted for use at high temperatures. In one industrial procedure, specimens of magnesia (basic) refractory are fed individually from a magazine into a three-point loading zone at the and is known as the stress tensor
Mechanical behaviour of materials 201 35 r- : 30 ,rx 25 ~ 20 15 Carburized i c' - C)~...... ~.004% Decarburized _ _ i _ i i ..... I I 104 I 0 S I 06 107 a 08 Cycles ~ N 250 200 E z if) L 150 m 100 Figure 7.5 S-N curve for carburized and decarburized iron. ~F ,,, [-i,- ....... a r~-/,,a d I , 2 2 3 FL MoR- 2 bd 2 I 32P0~nt bend I F F - | 2 2 3 F (L - L,) MoR= 2 bd 2 I 4iPo'nibend i Figure 7.6 Bend test configurations. MoR = modulus of rupture, F = applied force, L = outer span, Li = inner span, b = breadth of specimen, d = depth of specimen. known as the modulus of rupture (MoR) and expresses the maximum tensile stress which develops on the convex face of the loaded beam. Strong ceramics, such as silicon carbide and hot-pressed silicon nitride, have very high MoR values. The four-point loading method is often preferred because it subjects a greater volume and area of the beam to stress and is therefore more searching. MoR values from four-point tests are often substantially lower than those from three-point tests on the same material. Similarly, strength values tend to decrease as the specimen size is increased. To provide worthwhile data for quality control and design activities, close attention must be paid to strain rate and environment, and to the size, edge finish and surface texture of the specimen. With oxide ceramics and silica glasses, a high strain rate will give an appreciably higher flexural strength value than a low strain rate, which leads to slow crack growth and delayed fracture (Section 10.7). The bend test has also been adapted for use at high temperatures. In one industrial procedure, specimens of magnesia (basic) refractory are fed individually from a magazine into a three-point loading zone at the centre of an electric furnace heated by SiC elements. A similar type of hot-bend test has been used for the routine testing of graphite electrode samples and gives a useful indication of their ability to withstand accidental lateral impact during service in steel melting furnaces. Proof-testing is a long-established method of testing certain engineering components and structures. In a typical proof test, each component is held at a certain proof stress for a fixed period of time; loading and unloading conditions are standardized. In the case of ceramics, it may involve bend-testing, internal pressurization (for tubes) or rotation at high speed ('overspeeding' of grinding wheels). Components that withstand the proof test are, in the simplest analysis, judged to be sound and suitable for long-term service at the lower design stress. The underlying philosophy has been often questioned, not least because there is a risk that the proof test itself may cause incipient cracking. Nevertheless, proof-testing now has an important role in the statistical control of strength in ceramics. 7.2 Elastic deformation 7.2.1 Elastic deformation of metals It is well known that metals deform both elastically and plastically. Elastic deformation takes place at low stresses and has three main characteristics, namely (1) it is reversible, (2)stress and strain are linearly proportional to each other according to Hooke's Law and (3) it is usually small (i.e. <1% elastic strain). The stress at a point in a body is usually defined by considering an infinitesimal cube surrounding that point and the forces applied to the faces of the cube by the surrounding material. These forces may be resolved into components parallel to the cube edges and when divided by the area of a face give the nine stress components shown in Figure 7.7. A given component cr~j is the force acting in the j-direction per unit area of face normal to the /-direction. Clearly, when i = j we have normal stress components (e.g. crx~) which may be either tensile (conventionally positive) or compressive (negative), and when i ~ j (e.g. r the stress components are shear. These shear stresses exert couples on the cube and to prevent rotation of the cube the couples on opposite faces must balance and hence o" U = crji. 1 Thus, stress has only six independent components. When a body is strained, small elements in that body are displaced. If the initial position of an element 1The nine components of stress tTij form a second-rank tensor usually written tTxx O'x y tTx z tY y x tT y y tY y z tYzx tTzy tYZZ and is known as the stress tensor